Broken Morse trajectories

Let $(f,X)$ be a Morse--Smale pair on a closed manifold $M$ (Morse--Smale pairs) and let $p,q$ be critical points. A broken Morse trajectory from $p$ to $q$ is a finite sequence $({\gamma }_1,\dots ,{\gamma }_r)$, $r\ge 1$, together with critical points $p=p_0,p_1,\dots ,p_r=q$, such that ${\gamma }_i\in \tilde{\mathcal{M}}(p_{i-1},p_i)$ for every $i$ (Parametrized Morse trajectory space); $r$ is its length and $p_1,\dots ,p_{r-1}$ its intermediate critical points. The set of all broken trajectories from $p$ to $q$ is written $\bar{\mathcal{M}}(p,q)$; a broken trajectory of length $r=1$ is an ordinary trajectory, so $\mathcal{M}(p,q)\subseteq \bar{\mathcal{M}}(p,q)$ under the orbit-set identification (Unparametrized Morse trajectory moduli space), and a broken trajectory of length $r=2$ is called once-broken. The components are nonconstant, the intermediate points strictly decrease in value and in index, and $r\le \lambda (p)-\lambda (q)$, by Broken Morse trajectories have strictly decreasing critical values and indices; in particular $p_i\mathrm{\ne }{p}_{i+1}$ is automatic, and $\bar{\mathcal{M}}(p,q)=\mathcal{M}(p,q)$ whenever $\lambda (p)-\lambda (q)=1$.

Two strings that differ only by a time translation of one of the components represent the same broken trajectory: each ${\gamma }_i$ is a parametrized representative of a point of the orbit set $\mathcal{M}(p_{i-1},p_i)$, and $\bar{\mathcal{M}}(p,q)$ is the set of strings of such orbit classes, read through the orbit-set identification of Unparametrized Morse trajectory moduli space. The published symbol $\tilde{\mathcal{M}}(p_{i-1},p_i)$ is defined for distinct critical points; consecutive critical points of a broken trajectory are distinct precisely because every component is nonconstant.

The strict decrease is the published statement of Broken Morse trajectories have strictly decreasing critical values and indices applied to the string of nonconstant components; a single nonconstant component drops the index by at least one, which is also the content of No Morse--Smale trajectories for nonpositive index drop, and iterating the drops gives $r\le \lambda (p)-\lambda (q)$. Here $\lambda$ is the Morse index of Nondegenerate critical points, nullity, index, and coindex. If $\lambda (p)\le \lambda (q)$ no string of nonconstant components can exist, so $\bar{\mathcal{M}}(p,q)=\mathrm{\varnothing }$ — in particular the case $p=q$ is empty — and for $\lambda (p)-\lambda (q)=1$ the only possible length is $r=1$, whence $\bar{\mathcal{M}}(p,q)=\mathcal{M}(p,q)$. No compactness, finiteness, orientation or topology is asserted here.